When Cauchy and Hölder Met Minkowski: A Tour through Well-Known Inequalities

Many classical inequalities are just statements about the convexity or concavity of certain (hidden) underlying functions. This is nicely illustrated by Hardy, Littlewood, and Pólya [5] whose Chapter III deals with “Mean values with an arbitrary function and the theory of convex functions,” and by Steele [12] whose Chapter 6 is called “Convexity—The third pillar.” Yet another illustration is the following proof of the arithmetic-mean-geometric-mean inequality (which goes back to Pólya). The inequality states that the arithmetic mean of n positive real numbers a1, . . . , an is always greater or equal to their geometric mean: